About the IODS project

I heard about this course in the MBDP Metagenomics course last spring. I wish to develop my R skills further and learn more about RMarkdown, version control and using GitHub.

1 Link to my GitHub repository

2 or: https://github.com/mnpuputti/IODS-project

IODS week 2. Regression and model validation: Data analysis

This week we practised data wrangling and data visualization for linear models in DataCamp with packages dplyr and ggplot2. We practised making scripts for data wrangling in R with “Learning2014” dataset. We used this dataset, modifed the data to match IODS-course needs, and we will call that part of the dataset “Students2014”. Dataset is available here.

Bringing data into R

Explore the “Students2014” dataset. This dataset contains part of the “the international survey of Approaches to Learning”

#Read data "students2014" into R from .csv-file created in the data wrangling part of week2.
students2014 <- read.csv("~/Documents/MAIJA/R_IODS/IODS-project/data/learning2014.csv", row.names = 1)
#Check dimensions and structure of the dataset.
dim(students2014)
## [1] 166   7
str(students2014)
## 'data.frame':    166 obs. of  7 variables:
##  $ gender  : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
##  $ age     : int  53 55 49 53 49 38 50 37 37 42 ...
##  $ attitude: int  37 31 25 35 37 38 35 29 38 21 ...
##  $ deep    : num  3.58 2.92 3.5 3.5 3.67 ...
##  $ stra    : num  3.38 2.75 3.62 3.12 3.62 ...
##  $ surf    : num  2.58 3.17 2.25 2.25 2.83 ...
##  $ points  : int  25 12 24 10 22 21 21 31 24 26 ...

There are 166 observations and seven different variables including: gender, age, attitude, deep, stra, surf and points. There are three different types of learning technique questions: “deep” = deep learning,“stra” = strategic learning and “surf” = surface learning. In addition there were categories for global attitude towards statistics “attitude” , exam points “points” and also the “gender” of the survey participants.

Load the libraries needed in data visualization

library(ggplot2)
library(dplyr)
library(GGally)

Check the data through visualizations

###Check graphical overview with basic plot of pairwise correlations
p1 <- plot(students2014, main = "Graphical overview of the students2014 dataset")

##Check graphical overview with finer, more informative visual presentation of pairwise correlation plot, with ggpairs
ggpairs(students2014, mapping = aes(col = gender, alpha=0.3), lower = list(combo = wrap("facethist", bins = 20)))

Summaries of the variables

###Check summaries of the different variables available 
summary(students2014)
##  gender       age           attitude          deep            stra      
##  F:110   Min.   :17.00   Min.   :14.00   Min.   :1.583   Min.   :1.250  
##  M: 56   1st Qu.:21.00   1st Qu.:26.00   1st Qu.:3.333   1st Qu.:2.625  
##          Median :22.00   Median :32.00   Median :3.667   Median :3.188  
##          Mean   :25.51   Mean   :31.43   Mean   :3.680   Mean   :3.121  
##          3rd Qu.:27.00   3rd Qu.:37.00   3rd Qu.:4.083   3rd Qu.:3.625  
##          Max.   :55.00   Max.   :50.00   Max.   :4.917   Max.   :5.000  
##       surf           points     
##  Min.   :1.583   Min.   : 7.00  
##  1st Qu.:2.417   1st Qu.:19.00  
##  Median :2.833   Median :23.00  
##  Mean   :2.787   Mean   :22.72  
##  3rd Qu.:3.167   3rd Qu.:27.75  
##  Max.   :4.333   Max.   :33.00
#or summaries could be also individually picked e.g.
summary(students2014$gender)
##   F   M 
## 110  56

From the graphical overview and summary we can see that there are more female participants in the gender category. Female participants are younger than male participants. Exam point means are on the similar level for both male and female participants. Male participants score better for the attitude points. There are no major correlations between any categories for either female or male participants.
Highest correlation with the “point” variable is observed with the variable “attitude” and “stra” and negative correlation *with “surf” variable“.* The mean age is 25.51 with range from 17.00 to 55.00 years. The mean observed exam points is 22.72 with minimum being 7.00 and maximum points being 33.00. Deep learning mean is 3.68, strategic learning mean is 3.121 and surface learning techniques mean is 2.787.

Regression model

#Create a regression model with three explanatory variables identified from pairwise correlation plot:stra + age + attitude,
model1 <- lm(points ~ stra + age + attitude, data = students2014)

#Check the summary of the regression model and plot the results
set.seed(123)
summary(model1)
## 
## Call:
## lm(formula = points ~ stra + age + attitude, data = students2014)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -18.1149  -3.2003   0.3303   3.4129  10.7599 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 10.89543    2.64834   4.114 6.17e-05 ***
## stra         1.00371    0.53434   1.878   0.0621 .  
## age         -0.08822    0.05302  -1.664   0.0981 .  
## attitude     0.34808    0.05622   6.191 4.72e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.26 on 162 degrees of freedom
## Multiple R-squared:  0.2182, Adjusted R-squared:  0.2037 
## F-statistic: 15.07 on 3 and 162 DF,  p-value: 1.07e-08

Summary shows that only “attitude” of the tested explanatory variables could be considered statistically significant based on the p-values. Attitude has the most significant impact on the points with positive p.value 4.72e-09, while stra is showing p-value 0.0621 and age 0.0981. Attitude shows estimate of 0.34, while estimate for stra is 1.0. For age variable the estimate is negative. We can continue with the “attitude” variable and re-fit the model without the non-significant variables.

#Re-fit the model with significant variable "attitude"
model2 <- lm(points ~ attitude, data = students2014)

#Check the summary of the regression model and plot the results
set.seed(123)
summary(model2)
## 
## Call:
## lm(formula = points ~ attitude, data = students2014)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -16.9763  -3.2119   0.4339   4.1534  10.6645 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 11.63715    1.83035   6.358 1.95e-09 ***
## attitude     0.35255    0.05674   6.214 4.12e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared:  0.1906, Adjusted R-squared:  0.1856 
## F-statistic: 38.61 on 1 and 164 DF,  p-value: 4.119e-09

Now we can evaluate, whether the model and the data fit. The R-squared values tell us how close to the fitted (multiple) regression line our data is located. Multiple R-squared is 0.1906. This means that our model explains ~ 19 % of the variability around its mean. Estimates for the parametre for the model is 0.35255 for the attitude with standard error of 0.05674.

#Check how does the model looks with "points and "attitude"
qplot(attitude, points, data = students2014) + geom_smooth(method = "lm")

#Draw and explore the diagnostic plots
par(mfrow = c(2,2))
plot(model2, which = c(1,2,5))

Residuals are the errors in modelling the target variable. We want to minimize the modelling errors. Residuals vs fitted plot: There cannot be detected major spreading of the values. However, few outliers can be detected when fitted values increase. Normal QQ-plot explores the normal distribution potential of the errors. The data fits the normal distribution reasonably well, except the few outliers at the lower quantiles. We can say that the errors are normally distributed in this model. Residuals vs Leverage plot shows that there are no clear individual values that impact the model.

IODS Week 3: Logistic regression

This dataset contain data on student achievements in two Portuguese schools. We look at how alcohol consumption habits effects student performance.

Dataset student.zip-file can be found here. and information on dataset background here.

Let’s load needed libraries

library(dplyr)
library(ggplot2)
library(tidyr)

Reading in the data

Let’s read in the data from our data wrangling part. Alternatively, the modified dataset can be downloaded from here

alc <- read.csv("~/Documents/MAIJA/R_IODS/IODS-project/data/alc.csv", row.names = 1)

#Let's check the variables:
names(alc)  
##  [1] "school"     "sex"        "age"        "address"    "famsize"   
##  [6] "Pstatus"    "Medu"       "Fedu"       "Mjob"       "Fjob"      
## [11] "reason"     "nursery"    "internet"   "guardian"   "traveltime"
## [16] "studytime"  "failures"   "schoolsup"  "famsup"     "paid"      
## [21] "activities" "higher"     "romantic"   "famrel"     "freetime"  
## [26] "goout"      "Dalc"       "Walc"       "health"     "absences"  
## [31] "G1"         "G2"         "G3"         "alc_use"    "high_use"
#or use glimpse
glimpse(alc)
## Observations: 382
## Variables: 35
## $ school     <fct> GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, G…
## $ sex        <fct> F, F, F, F, F, M, M, F, M, M, F, F, M, M, M, F, F, F,…
## $ age        <int> 18, 17, 15, 15, 16, 16, 16, 17, 15, 15, 15, 15, 15, 1…
## $ address    <fct> U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U,…
## $ famsize    <fct> GT3, GT3, LE3, GT3, GT3, LE3, LE3, GT3, LE3, GT3, GT3…
## $ Pstatus    <fct> A, T, T, T, T, T, T, A, A, T, T, T, T, T, A, T, T, T,…
## $ Medu       <int> 4, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 2, 4, 4, 2, 4, 4, 3,…
## $ Fedu       <int> 4, 1, 1, 2, 3, 3, 2, 4, 2, 4, 4, 1, 4, 3, 2, 4, 4, 3,…
## $ Mjob       <fct> at_home, at_home, at_home, health, other, services, o…
## $ Fjob       <fct> teacher, other, other, services, other, other, other,…
## $ reason     <fct> course, course, other, home, home, reputation, home, …
## $ nursery    <fct> yes, no, yes, yes, yes, yes, yes, yes, yes, yes, yes,…
## $ internet   <fct> no, yes, yes, yes, no, yes, yes, no, yes, yes, yes, y…
## $ guardian   <fct> mother, father, mother, mother, father, mother, mothe…
## $ traveltime <int> 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3,…
## $ studytime  <int> 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 1, 3, 2,…
## $ failures   <int> 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ schoolsup  <fct> yes, no, yes, no, no, no, no, yes, no, no, no, no, no…
## $ famsup     <fct> no, yes, no, yes, yes, yes, no, yes, yes, yes, yes, y…
## $ paid       <fct> no, no, yes, yes, yes, yes, no, no, yes, yes, yes, no…
## $ activities <fct> no, no, no, yes, no, yes, no, no, no, yes, no, yes, y…
## $ higher     <fct> yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes…
## $ romantic   <fct> no, no, no, yes, no, no, no, no, no, no, no, no, no, …
## $ famrel     <int> 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 3, 5, 4, 5, 4, 4, 3, 5,…
## $ freetime   <int> 3, 3, 3, 2, 3, 4, 4, 1, 2, 5, 3, 2, 3, 4, 5, 4, 2, 3,…
## $ goout      <int> 4, 3, 2, 2, 2, 2, 4, 4, 2, 1, 3, 2, 3, 3, 2, 4, 3, 2,…
## $ Dalc       <int> 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,…
## $ Walc       <int> 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, 1,…
## $ health     <int> 3, 3, 3, 5, 5, 5, 3, 1, 1, 5, 2, 4, 5, 3, 3, 2, 2, 4,…
## $ absences   <int> 5, 3, 8, 1, 2, 8, 0, 4, 0, 0, 1, 2, 1, 1, 0, 5, 8, 3,…
## $ G1         <int> 2, 7, 10, 14, 8, 14, 12, 8, 16, 13, 12, 10, 13, 11, 1…
## $ G2         <int> 8, 8, 10, 14, 12, 14, 12, 9, 17, 14, 11, 12, 14, 11, …
## $ G3         <int> 8, 8, 11, 14, 12, 14, 12, 10, 18, 14, 12, 12, 13, 12,…
## $ alc_use    <dbl> 1.0, 1.0, 2.5, 1.0, 1.5, 1.5, 1.0, 1.0, 1.0, 1.0, 1.5…
## $ high_use   <lgl> FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE…

These parametres are fully explained at the data repository site. Key variables explained shortly: Dalc describes workday alcohol consumption, Walc weekend alcohol consumption, alc_use describes average alcohol use per day. All these three parametres have scale from 1 (very low) to 5 (very high use). High_use describes whether student uses alcohol more than twice a week.

Let’s see how the data looks in order to pick 4 interesting variables:

# draw a bar plot of each variable
gather(alc) %>% ggplot(aes(value)) + geom_bar(color="white",fill="pink") + facet_wrap("key", scales = "free")

Let’s pick variables: absences, school, age and sex and explore how they correlate with alcohol consumption. Our hypothesis is that males have higher rate in high use of alcohol than females. We can also hypothesize that high use of alcohol leads to more absences from school. We can test whether high alcohol use is linked to age. And we can see if there is a difference in culture towards alcohol use in these two different schools.

First, Let’s explore how students use alcohol in these two schools and whether sex can be associated with it.

plot1 <- ggplot(data = alc, aes(x=alc_use, fill=school))
plot1 + geom_bar() +
  facet_wrap("sex")

Here we can see that school “GP” shows higher average counts for alcohol use compared to school “MS”. In addition, the female participants have higher counts of very low or low average alcohol consumption compared to males.

Let’s see whether this changes for high_use

plot2 <- ggplot(data = alc, aes(x=high_use, fill=school))
plot2 + geom_bar() +
  facet_wrap("sex")

Also higher counts for high_alcohol consumption in school “GP” and male students. This confirms our hypothesis that male students consume more alcohol and that students in different schools may have different drinking culture and attitude towards drinking alcohol.

Let’s see what is the relationship with absences, high_alc usage and sex.

# plot1 explores the relationship of high_use and absences
plot3 <- ggplot(alc, aes(x = high_use, y = absences, col = sex))

# define the plot as a boxplot and draw it
plot3 + geom_boxplot() + ylab("absences") + ggtitle("Student absences by high alcohol consumption")

Here we can see that high use of alcohol increases the number of absences. Male students have more absences compared to female students. That is expected, since we already observed that male students have higher average alcohol usage.

# plot1 explores the relationship of high_use and absences with  variable age
plot4 <- ggplot(alc, aes(x = high_use, y = absences))

# define the plot as a boxplot and draw it
plot4 + geom_boxplot(color="darkblue") + ylab("absences") + facet_wrap("age") + ggtitle("Student absences by high alcohol consumption and age") + theme_bw()

Here we can see that high alcohol consumption is most common for 17 year old students and that they increase the absences.

Logistic regression

Let’s explore how high use of alcohol is linked to explanatory variables and use logistic regression

#We can now explore the relationship between four variables and high alcohol consumption with logistic regression
model1 <- glm(high_use ~ school+absences+sex+age, data = alc, family = "binomial")

#Let's study the summary of our fitted model
summary(model1)
## 
## Call:
## glm(formula = high_use ~ school + absences + sex + age, family = "binomial", 
##     data = alc)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.3081  -0.8357  -0.6384   1.0811   2.1077  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -4.29534    1.79304  -2.396   0.0166 *  
## schoolMS     0.30068    0.39803   0.755   0.4500    
## absences     0.09467    0.02342   4.042 5.31e-05 ***
## sexM         0.99332    0.24134   4.116 3.86e-05 ***
## age          0.14593    0.10850   1.345   0.1786    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 465.68  on 381  degrees of freedom
## Residual deviance: 426.48  on 377  degrees of freedom
## AIC: 436.48
## 
## Number of Fisher Scoring iterations: 4

Here we can see that age and school are not statistically significant for high_use of alcohol despite our previous interpretation through visualization. However, we have factor variables here for school and sex and they are demonstrated in the intercept. Let’s explore the model without school variable.

#Let's re-fit our model, add -1 to exclude interception and show coefficient estimate values straight
model2 <- glm(high_use ~ sex+age+absences, data = alc, family = "binomial")

#Let's study the summary of our fitted model
summary(model2)
## 
## Call:
## glm(formula = high_use ~ sex + age + absences, family = "binomial", 
##     data = alc)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.3002  -0.8428  -0.6386   1.0781   2.1068  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -4.73160    1.69730  -2.788  0.00531 ** 
## sexM         0.98440    0.24074   4.089 4.33e-05 ***
## age          0.17516    0.10125   1.730  0.08365 .  
## absences     0.09223    0.02314   3.986 6.71e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 465.68  on 381  degrees of freedom
## Residual deviance: 427.04  on 378  degrees of freedom
## AIC: 435.04
## 
## Number of Fisher Scoring iterations: 4

Now we can see that coefficient for female students is seen in the intercept: -4.7 and for male student the estimate changed and is -0.98

#Let's study the coefficients of our model
coef(model2)
## (Intercept)        sexM         age    absences 
## -4.73159988  0.98439673  0.17515570  0.09223477
#Next we can explore the coefficients and their confidence intervals as odd ratios 
# We can compute odds ratios (oddrat)
oddrat <- coef(model2) %>% exp

# compute confidence intervals (confin)
confin <- confint(model2) %>% exp

#Npw we can print out our model's coefficients oddratios with their confidence intervals and join them
cbind(oddrat, confin)
##                  oddrat        2.5 %    97.5 %
## (Intercept) 0.008812361 0.0002990615 0.2356076
## sexM        2.676196933 1.6793970873 4.3232945
## age         1.191431709 0.9782468520 1.4562441
## absences    1.096622249 1.0500945691 1.1499088

Odd ratio describes the ratio of successes to failures. Here we can see that odds ratio for all explanatory variables is higher than 1 for all explanatory variables. This means that e.g. male students have 2.7 times probability to use high levels of alcohol than female students. Absences are likely to increase odds for high alcohol consumption.

Predictions

Absences and sex were statistically significant as explanatory variables. We can now re-fit our model with these variables and explore the predictions.

# re-fit the model with statistically significant variables
model3 <- glm(high_use ~ absences + sex, data = alc, family = "binomial")

# predict() the probability of high_use
probabilities <- predict(model3, type = "response")

# add the predicted probabilities to 'alc'
alc <- mutate(alc, probability = probabilities)

# use the probabilities to make a prediction of high_use
alc <- mutate(alc, prediction = probabilities>0.5)


# see the last ten original classes, predicted probabilities, and class predictions
select(alc, absences, sex, high_use, probability, prediction) %>%  tail(10) 
##     absences sex high_use probability prediction
## 373        0   M    FALSE   0.2976656      FALSE
## 374        7   M     TRUE   0.4545495      FALSE
## 375        1   F    FALSE   0.1493808      FALSE
## 376        6   F    FALSE   0.2215747      FALSE
## 377        2   F    FALSE   0.1620742      FALSE
## 378        2   F    FALSE   0.1620742      FALSE
## 379        2   F    FALSE   0.1620742      FALSE
## 380        3   F    FALSE   0.1756235      FALSE
## 381        4   M     TRUE   0.3841248      FALSE
## 382        2   M     TRUE   0.3395595      FALSE

Here we can see that all our predictions remain false and true cases of high_use are not predicted correctly. Let’s further explore this visually.

# tabulate the target variable versus the predictions
table(high_use = alc$high_use, prediction = alc$prediction)
##         prediction
## high_use FALSE TRUE
##    FALSE   258   10
##    TRUE     88   26

There were 88 false predictions of actual high_use and 26 accurate predictions of high_use.

# initialize a plot of 'high_use' versus 'probability' in 'alc'
g <- ggplot(alc, aes(x = probability, y = high_use, col = prediction))

# define the geom as points and draw the plot
g + geom_point()

Here we can see that the rate for true prediction increases, when probability reaches 0.5.

# tabulate the target variable versus the predictions
table(high_use = alc$high_use, prediction = alc$prediction) %>% prop.table %>% addmargins 
##         prediction
## high_use      FALSE       TRUE        Sum
##    FALSE 0.67539267 0.02617801 0.70157068
##    TRUE  0.23036649 0.06806283 0.29842932
##    Sum   0.90575916 0.09424084 1.00000000
Bonus: Exploration of cross-validation and wrong predictions
# the logistic regression model m and dataset alc (with predictions) are available

# define a loss function (average prediction error)
loss_func <- function(class, prob) {
  n_wrong <- abs(class - prob) > 0.5
  mean(n_wrong)
}

# compute the average number of wrong predictions in the (training) data


# K-fold cross-validation
library(boot)
crossval1 <- cv.glm(data = alc, cost = loss_func, glmfit = model3, K = 10)

# average number of wrong predictions in the cross validation
crossval1$delta[1]
## [1] 0.2670157

Since some of the non-significant variables were already excluded from our final model here, we explored the same variables as in DataCamp and model reached around the same 0.26 error.

Super-Bonus: exploration of different model options and their error rate
#Let's explore our previous models model 1 and model 2.
# K-fold cross-validation
#model 1
crossval2 <- cv.glm(data = alc, cost = loss_func, glmfit = model1, K = 10)

# average number of wrong predictions in the cross validation
crossval2$delta[1]
## [1] 0.2434555
# K-fold cross-validation
#model 2
crossval3 <- cv.glm(data = alc, cost = loss_func, glmfit = model2, K = 10)

# average number of wrong predictions in the cross validation
crossval3$delta[1]
## [1] 0.2486911
############# 
#And let's try a new test model to explore other variable options internet, school, age and failures
model4 <- glm(high_use ~ internet+school+age+failures, data = alc, family = "binomial")

# K-fold cross-validation
#model 4
crossval4 <- cv.glm(data = alc, cost = loss_func, glmfit = model4, K = 10)

# average number of wrong predictions in the cross validation
crossval4$delta[1]
## [1] 0.3062827
#Let's remove some of the variables one by one to see the effect on predictiosn
model5 <- glm(high_use ~ internet+age+failures, data = alc, family = "binomial")

# K-fold cross-validation
#model 4
crossval5 <- cv.glm(data = alc, cost = loss_func, glmfit = model4, K = 10)

# average number of wrong predictions in the cross validation
crossval5$delta[1]
## [1] 0.3036649
#And let's remove more variables:
model6 <- glm(high_use ~ internet+failures, data = alc, family = "binomial")

# K-fold cross-validation
#model 4
crossval6 <- cv.glm(data = alc, cost = loss_func, glmfit = model4, K = 10)

# average number of wrong predictions in the cross validation
crossval6$delta[1]
## [1] 0.3010471
#And let's try a new model to explore options:
model7 <- glm(high_use ~ internet, data = alc, family = "binomial")

# K-fold cross-validation
#model 4
crossval7 <- cv.glm(data = alc, cost = loss_func, glmfit = model4, K = 10)

# average number of wrong predictions in the cross validation
crossval7$delta[1]
## [1] 0.2984293
#And let's try a new model to explore options:
model7 <- glm(high_use ~ failures, data = alc, family = "binomial")

# K-fold cross-validation
#model 4
crossval7 <- cv.glm(data = alc, cost = loss_func, glmfit = model4, K = 10)

# average number of wrong predictions in the cross validation
crossval7$delta[1]
## [1] 0.3141361

Here we can see, that the test model 4 had increased prediction error (31 %) compared to previous model. Model 4 combined internet access, age, school and failures in our new test model. All the remaining attempt by removing one variable at the time did not decrease the prediction error.

IODS Week 4: Clustering and classification

Reading in the Boston dataset included in the MASS package and load in other needed libraries. Boston dataset contains Housing related data from Boston suburbs, and we explore the town crime rates with the explanatory variables from this dataset.

library(MASS)
library(tidyverse)
library(corrplot)
library(viridis)
library(ggplot2)
library(corrplot)
data(Boston)

Let’s explore the Boston dataset

str(Boston)
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
dim (Boston)
## [1] 506  14

Boston dataset has 506 observations and 14 different variables. More information on different variables is available here.

#Let's explore the Boston data with graphical overview and summaries of the variables
pairs(Boston)

summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08204   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00

From the summary we can see e.g. the mean crime rate in Boston suburbs is 3.6 crimes/capita. nox tells us the nitrogen oxide concentrations (ppm) with mean value in Boston of 0.6. dis tells us that the distance from the Boston employment centres ranges from 1.1 to 12.

Correlation plot
#or exploration with corrplot
# First we need to calculate the correlation matrix, optionally print it and round it
cor_matrix<-cor(Boston) 
#cor_matrix
roudedcormatrix<- cor_matrix %>% round(digits = 2) 

#plot 
corrplot(roudedcormatrix, type = "upper", cl.pos = "b", tl.pos = "d", tl.cex = 0.6)

From the correlation plot we can see the positive and negative correlations with different variables and crime rate (crim). Crime rate has the highest positive correlation with variables rad (index of accessibility to radial highways) and tax (full-value property-tax rate per $10,000). On the opposite, lowest correlations are detected with medv (median value of owner-occupied homes in $1000s), black (1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town) and dis (weighted mean of distances to five Boston employment centres).

Scaling
#Let's standardise our dataset and scale our data
Bostonscaled <- scale(Boston)

#Let's change our matrix into a dataframe 
Bostonscaled<-as.data.frame(Bostonscaled)

#Check the class 
class(Bostonscaled)
## [1] "data.frame"
#Let's check the summary
summary(Bostonscaled)
##       crim                 zn               indus        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202  
##       chas              nox                rm               age         
##  Min.   :-0.2723   Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331  
##  1st Qu.:-0.2723   1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366  
##  Median :-0.2723   Median :-0.1441   Median :-0.1084   Median : 0.3171  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.:-0.2723   3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059  
##  Max.   : 3.6648   Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164  
##       dis               rad               tax             ptratio       
##  Min.   :-1.2658   Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047  
##  1st Qu.:-0.8049   1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876  
##  Median :-0.2790   Median :-0.5225   Median :-0.4642   Median : 0.2746  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6617   3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058  
##  Max.   : 3.9566   Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372  
##      black             lstat              medv        
##  Min.   :-3.9033   Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.: 0.2049   1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median : 0.3808   Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4332   3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 0.4406   Max.   : 3.5453   Max.   : 2.9865
#Let's create quantiles for the crime rates 
crimequantiles<- quantile(Bostonscaled$crim)
crimequantiles
##           0%          25%          50%          75%         100% 
## -0.419366929 -0.410563278 -0.390280295  0.007389247  9.924109610
#and rename them with labelstring
labelstring<-c("low","med_low","med_high","high")
crime <- cut(Bostonscaled$crim, breaks = crimequantiles, include.lowest = TRUE, label = labelstring)

#Let's make changes into the dataset and remove original crim
Bostonscaled <- dplyr::select(Bostonscaled, -crim)

# add the new categorical value to scaled data
Bostonscaled <- data.frame(Bostonscaled, crime)
Now we can divide the dataset into test and train sets
#Let's see the number of rows in our dataset
n <- nrow(Bostonscaled)

#We can choose randomly 80% of the rows
ind <- sample(n,  size = n * 0.8)

# We can create train set
train <- Bostonscaled[ind,]

# We can create test set 
test <- Bostonscaled[-ind,]

Linear discriminant analysis (LDA)

#We can now use the categorical crime rate as the target variable and all the other variables in the dataset as predictor variables (as dot in script)
lda.fit <- lda(crime ~.,data = train)

lda.fit
## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2698020 0.2202970 0.2574257 0.2524752 
## 
## Group means:
##                   zn      indus        chas        nox          rm
## low       0.94776710 -0.8971760 -0.09172814 -0.8661508  0.41403330
## med_low  -0.09773272 -0.2625464  0.08156758 -0.5620961 -0.13125022
## med_high -0.41220520  0.2703592  0.25766519  0.4240806 -0.00487157
## high     -0.48724019  1.0171096 -0.04073494  1.0406599 -0.38642231
##                 age        dis        rad        tax     ptratio
## low      -0.8731314  0.8574965 -0.6984422 -0.7588152 -0.40831234
## med_low  -0.3444234  0.3681568 -0.5534543 -0.4905469 -0.06872784
## med_high  0.4573932 -0.4064705 -0.3999076 -0.2478149 -0.20286305
## high      0.8080500 -0.8529113  1.6382099  1.5141140  0.78087177
##                black       lstat         medv
## low       0.38256024 -0.76416040  0.499383470
## med_low   0.35237238 -0.11838895  0.009138486
## med_high  0.08931233  0.08857418  0.056652597
## high     -0.84222907  0.83994701 -0.633454131
## 
## Coefficients of linear discriminants:
##                 LD1         LD2         LD3
## zn       0.10049915  0.72763600 -0.93452410
## indus   -0.06232910 -0.25446786  0.46592588
## chas    -0.06856708 -0.09745969  0.11177269
## nox      0.35900138 -0.52823345 -1.38164587
## rm      -0.10933789 -0.06468908 -0.08375578
## age      0.32618623 -0.34883089 -0.24539175
## dis     -0.11348351 -0.22009469  0.31728047
## rad      2.91204021  1.07347198  0.18122290
## tax      0.11480221 -0.22548808  0.37179438
## ptratio  0.10464710  0.05358570 -0.31555558
## black   -0.15215701 -0.01281596  0.17254010
## lstat    0.16466416 -0.22434855  0.48464256
## medv     0.14840845 -0.29710857 -0.12316117
## 
## Proportion of trace:
##   LD1   LD2   LD3 
## 0.944 0.043 0.013

From the summary of LDA we can see that LDA model explains over 90 % of the variance. In addition, we can see the different explanatory variable means, and we can check how they vary between crime rates, e.g. age, nox and dis show variation between low crime and high crime rate means.

LDA biplot
# Let's create the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "color", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

# Set target classes as numeric
classes <- as.numeric(train$crime)

# Now we can biplot the LDA results
p <- plot(lda.fit, dimen = 2, col = classes, pch = classes) 

p + lda.arrows(lda.fit, col = classes, myscale = 1.5) 

## integer(0)

Or We can also try ggplot2 and viridis packages for LDA biplot visualization Example script can be found here and more on data management prior to ggplotting can be found here

require(scales)
require(gridExtra)

#We can also try ggplot2 and viridis packages for LDA biplot visualization 

r <- lda(formula = crime ~ ., 
         data = train, 
         prior = c(1,1,1,1)/4)

prop = r$svd^2/sum(r$svd^2)

##
lda <- lda(crime ~ .,data=train, prior = c(1,1,1,1)/4)

prop.lda = r$svd^2/sum(r$svd^2)

plda <- predict(object = lda, newdata = train)


dataset2 = data.frame(crime = train[,"crime"], lda = plda$x)

p1 <- ggplot(dataset2) + geom_point(aes(lda.LD1, lda.LD2, colour = crime), size = 2.5) + 
  labs(x = paste("LD1 (", percent(prop.lda[1]), ")", sep=""),
y = paste("LD2 (", percent(prop.lda[2]), ")", sep="")) +
    scale_color_viridis(discrete = TRUE, option = "plasma") +
  theme_bw()

p1

We can also try ordinations in ggord instead.

#library(devtools)
#install_github("fawda123/ggord")
library(ggord)

ord <- lda(crime ~ ., train, color = crime, prior = rep(1, 4)/4)
ggord(ord, train$crime, color = crime, arrow=0.6, txt=4, size=3)

From the plots we can see that only group clearly separating from other groups is the high crime rate group. In addition, we can see that the explanatory variable rad points out towards high crime rate and nox points out towards med_high crime rate, where as explanatory variable zn points out towards low crime rate. This means that rad, which was “index of accessibility to radial highways”, could almost solely separate the high crime rate grouping in this model.

library(dplyr)

# We can now save the new classes from test data
correct_classes <- test$crime
correct_classes
##   [1] low      med_low  med_low  med_high med_high med_high med_high
##   [8] med_low  med_low  med_low  med_low  med_low  med_low  med_low 
##  [15] med_low  low      med_low  low      low      med_low  med_low 
##  [22] med_low  med_low  med_low  med_low  med_low  med_low  med_low 
##  [29] med_low  med_low  med_high med_high med_high med_high med_high
##  [36] med_low  low      low      low      med_low  low      low     
##  [43] med_low  low      low      low      med_low  med_high med_high
##  [50] med_high med_high med_high med_high med_low  med_low  med_high
##  [57] med_high med_high med_high med_high med_high med_low  low     
##  [64] med_low  med_low  low      med_low  med_high med_low  low     
##  [71] low      low      low      med_low  high     high     high    
##  [78] high     high     high     high     high     high     high    
##  [85] high     high     high     high     high     high     high    
##  [92] high     high     high     high     high     high     high    
##  [99] high     med_low  med_low  med_low 
## Levels: low med_low med_high high
# And we can remove the crime variable from test data
test <- dplyr::select(test, -crime)

Predictions

#Predictions of classes with LDA model and test data
set.seed(123)
lda.pred <- predict(lda.fit, newdata = test)

# And we can then cross tabulate the results
predictions<-table(correct = correct_classes, predicted = lda.pred$class)
predictions
##           predicted
## correct    low med_low med_high high
##   low       12       5        1    0
##   med_low    7      18       12    0
##   med_high   1       7       14    0
##   high       0       0        0   25

Predictions with the test data show that 14 of low crime rate was correctly predicted as low and 5 falsely as med_low. In addition, 5 med_high crime rated was falsely predicted as low and 9 as med_high. From the high crime rate 31 was correctly predicted as high. The model seems to work well with the high crime rate predictions. However, there can be seen more false predictions for med_high, med_low and low crime rates.

NOTE! Despite the set.seed(), this table seems to change the values with every knit and I did not find a solution to this yet

Predicting no. of clusters

#We can reload the Boston dataset
data(Boston)

#Let's standardise the dataset by scaling the variables, 
#we need data.frame
Bostonscaled2 <- as.data.frame(Boston)

#Now we can use k-means clustering to calculate the distances
# k-means clustering
km <-kmeans(Bostonscaled2, centers = 4)
#km$cluster

#plot
pairs(Bostonscaled2[1:10], col = km$cluster)

####
set.seed(123)
# determine the number of clusters
k_max <- 10

# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(Bostonscaled2, k)$tot.withinss})

# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')

From the qplot we can see that the optimal number of clusters is 2, since there is a radical rise.

#Let's set the number of centers to 2 and cluster again.

# k-means clustering
km2 <-kmeans(Bostonscaled2, centers = 2)

# plot the Boston dataset with clusters
pairs(Bostonscaled2[1:10], col = km2$cluster)

When we compare the clustering of crime vs explanatory variables, good grouping can be assumed e.g. in rad and tax. In comparison, nox and rm scatter more.

#Bonus

#Superbonus

IODS week 5: Dimensionality reduction techniques

This week we explore the human dataset, which can be found here

In addition, related metadata is available here and here.

This dataset contains information on Human development projects by UN containing key aspect statistics such as life expectancy (Life.Exp) and maternal mortality rate (Mat.Mor).

Let’s read in the data

human<-read.csv("~/Documents/MAIJA/R_IODS/IODS-project/data/human.csv", row.names = 1)

Let’s check basic structure, dimensions and summary first.

str(human)
## 'data.frame':    155 obs. of  8 variables:
##  $ Edu2.FM  : num  1.007 0.997 0.983 0.989 0.969 ...
##  $ Labo.FM  : num  0.891 0.819 0.825 0.884 0.829 ...
##  $ Edu.Exp  : num  17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
##  $ Life.Exp : num  81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
##  $ GNI      : int  64992 42261 56431 44025 45435 43919 39568 52947 42155 32689 ...
##  $ Mat.Mor  : int  4 6 6 5 6 7 9 28 11 8 ...
##  $ Ado.Birth: num  7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
##  $ Parli.F  : num  39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...
dim(human)
## [1] 155   8
summary(human)
##     Edu2.FM          Labo.FM          Edu.Exp         Life.Exp    
##  Min.   :0.1717   Min.   :0.1857   Min.   : 5.40   Min.   :49.00  
##  1st Qu.:0.7264   1st Qu.:0.5984   1st Qu.:11.25   1st Qu.:66.30  
##  Median :0.9375   Median :0.7535   Median :13.50   Median :74.20  
##  Mean   :0.8529   Mean   :0.7074   Mean   :13.18   Mean   :71.65  
##  3rd Qu.:0.9968   3rd Qu.:0.8535   3rd Qu.:15.20   3rd Qu.:77.25  
##  Max.   :1.4967   Max.   :1.0380   Max.   :20.20   Max.   :83.50  
##       GNI            Mat.Mor         Ado.Birth         Parli.F     
##  Min.   :   581   Min.   :   1.0   Min.   :  0.60   Min.   : 0.00  
##  1st Qu.:  4198   1st Qu.:  11.5   1st Qu.: 12.65   1st Qu.:12.40  
##  Median : 12040   Median :  49.0   Median : 33.60   Median :19.30  
##  Mean   : 17628   Mean   : 149.1   Mean   : 47.16   Mean   :20.91  
##  3rd Qu.: 24512   3rd Qu.: 190.0   3rd Qu.: 71.95   3rd Qu.:27.95  
##  Max.   :123124   Max.   :1100.0   Max.   :204.80   Max.   :57.50

From this summary we can see that the Mat.Mor (maternal mortality rate) varies a lot between the countries with minimum value of 1 to up to 1100. The expected years of schooling ( Edu.Exp ) also varies a lot with mean of 13.18 years.

Then we can create a graphical overview of the data. This week we want to explore the data through dimension reduction techniques and we want to standardise our values. We can use package GGally for visualization with ggpairs.

library(GGally)
ggpairs(human)

Based on this visualization, we could evaluate that most variables of the human dataset does not seem to follow normal distribution. Here we can see that the highest positive correlation are between Life.Exp (Life expectancy at birth, years) and Edu.Exp (Expected years of schooling). In addition, high correlation is also detected between Mat.Mor (Maternal mortality rate) and Ado.Birth (Adolescent birth rate). In addition, there is a negative correlation between Life.Exp and Mat.Mor.

Principal component analysis (PCA)

No standardization

Let’s check also the summaries of the non-standardised data to explore the importance of different components.

pca_human <- prcomp(human)

summary_non <- summary(pca_human)
summary_non
## Importance of components:
##                              PC1      PC2   PC3   PC4   PC5   PC6    PC7
## Standard deviation     1.854e+04 185.5219 25.19 11.45 3.766 1.566 0.1912
## Proportion of Variance 9.999e-01   0.0001  0.00  0.00 0.000 0.000 0.0000
## Cumulative Proportion  9.999e-01   1.0000  1.00  1.00 1.000 1.000 1.0000
##                           PC8
## Standard deviation     0.1591
## Proportion of Variance 0.0000
## Cumulative Proportion  1.0000

Let’s see how principal component analysis (PCA) looks without standardization.

biplot(pca_human, choices = 1:2, cex = c(0.8, 1), col = c("grey30", "deeppink"))

#Let's add the percentages
# rounded percentages of variance captured by each PC
pca_pr<- round(100*summary_non$importance[2,], digits = 1) 

# print out the percentages of variance
pca_pr
## PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 
## 100   0   0   0   0   0   0   0
# create object pc_lab to be used as axis labels
pc_lab <- paste0(names(pca_pr), " (", pca_pr, "%)")

# draw a biplot
biplot(pca_human, cex = c(0.8, 1), col = c("grey30", "deeppink"), xlab = pc_lab[1], ylab = pc_lab[2])

Here we can see the PCA biplot with non-standardized data. Without standardization the PC1 explains 100 % of the variability and all the variance seems to be explained by one variable, GNI (gross national income, per capita). As we can see from the summary, the other PC components are not explaining any part of the variability.

Standardization

Now let’s repeat the PCA with standardised values. Let’s standardise the values with “scale”.

First, let’s check the summaries of the standardized data

human_std <- scale(human)

pca_human_std <- prcomp(human_std)

summary_std <- summary(pca_human_std)
summary_std
## Importance of components:
##                           PC1    PC2     PC3     PC4     PC5     PC6
## Standard deviation     2.0708 1.1397 0.87505 0.77886 0.66196 0.53631
## Proportion of Variance 0.5361 0.1624 0.09571 0.07583 0.05477 0.03595
## Cumulative Proportion  0.5361 0.6984 0.79413 0.86996 0.92473 0.96069
##                            PC7     PC8
## Standard deviation     0.45900 0.32224
## Proportion of Variance 0.02634 0.01298
## Cumulative Proportion  0.98702 1.00000
#We want to add the percentages to the biplot

# rounded percentages of variance captured by each PC
pca_pr_std <- round(100*summary_std$importance[2,], digits = 1) 

# print out the percentages of variance
pca_pr_std
##  PC1  PC2  PC3  PC4  PC5  PC6  PC7  PC8 
## 53.6 16.2  9.6  7.6  5.5  3.6  2.6  1.3
# create object pc_lab to be used as axis labels
pc_lab_std <- paste0(names(pca_pr_std), " (", pca_pr_std, "%)")

# draw a biplot
biplot(pca_human_std, cex = c(0.8, 1), col = c("grey30", "deeppink"), xlab = pc_lab_std[1], ylab = pc_lab_std[2])

Now we can see that the results between standardised and non-standardised PCA look very different.

Here we can see that 53.6 % of the variability is explained by the PC1 and 16.2 % by the PC2. We can see in the standardised PC biplot that that Labo.FM (labour market participation rate by sex) and Parli.F (share of females in parliament, percent) influence mostly to PC2 whereas e.g. Mat.Mor and Ado.Birth influence on PC1. Most Nordic and European countries are located to upper left side of the PC plot as are variables such as higher Life.Exp and higher GNI.

Standardization allowed us to be able to observe differences between countries by other variables than only GNI.

Multiple correspondence analysis (MCA)

Tea dataset

Let’s load in another dataset: tea from the package FactoMineR and explore the structure and dimensions of this dataset.

library(FactoMineR)

#Let's load the tea-dataset
data(tea)

str(tea)
## 'data.frame':    300 obs. of  36 variables:
##  $ breakfast       : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
##  $ tea.time        : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
##  $ evening         : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
##  $ lunch           : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
##  $ dinner          : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
##  $ always          : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
##  $ home            : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
##  $ work            : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
##  $ tearoom         : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
##  $ friends         : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
##  $ resto           : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
##  $ pub             : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
##  $ Tea             : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
##  $ How             : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
##  $ sugar           : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
##  $ how             : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ where           : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ price           : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
##  $ age             : int  39 45 47 23 48 21 37 36 40 37 ...
##  $ sex             : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
##  $ SPC             : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
##  $ Sport           : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
##  $ age_Q           : Factor w/ 5 levels "15-24","25-34",..: 3 4 4 1 4 1 3 3 3 3 ...
##  $ frequency       : Factor w/ 4 levels "1/day","1 to 2/week",..: 1 1 3 1 3 1 4 2 3 3 ...
##  $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
##  $ spirituality    : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
##  $ healthy         : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
##  $ diuretic        : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
##  $ friendliness    : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
##  $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
##  $ feminine        : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
##  $ sophisticated   : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
##  $ slimming        : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ exciting        : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
##  $ relaxing        : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
##  $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...
dim(tea)
## [1] 300  36
ggpairs(tea[1:5])

There were 300 different observations and 36 different variables. Here we explored visually only first five of them and yet the visualization does not look very good or informative. So we should try something else. This time we want to explore this dataset with multiple correspondence analysis (MCA). Let’s see how that looks.

First, let’s pick only a limited number of variables and explore the structure of this dataset

library(ggplot2)
library(tidyr)
library(dplyr)

# column names to keep in the dataset
keep_columns <- c("Tea", "How", "how", "sugar", "where", "sex", "lunch")

# select the 'keep_columns' to create a new dataset
tea_time <- dplyr::select(tea, one_of(keep_columns))

# look at the summaries and structure of the data
summary(tea_time)
##         Tea         How                      how           sugar    
##  black    : 74   alone:195   tea bag           :170   No.sugar:155  
##  Earl Grey:193   lemon: 33   tea bag+unpackaged: 94   sugar   :145  
##  green    : 33   milk : 63   unpackaged        : 36                 
##                  other:  9                                          
##                   where     sex           lunch    
##  chain store         :192   F:178   lunch    : 44  
##  chain store+tea shop: 78   M:122   Not.lunch:256  
##  tea shop            : 30                          
## 
str(tea_time)
## 'data.frame':    300 obs. of  7 variables:
##  $ Tea  : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
##  $ How  : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
##  $ how  : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ sugar: Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
##  $ where: Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ sex  : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
##  $ lunch: Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
# multiple correspondence analysis
mca <- MCA(tea_time, graph = FALSE)

# summary of the model
summary(mca)
## 
## Call:
## MCA(X = tea_time, graph = FALSE) 
## 
## 
## Eigenvalues
##                        Dim.1   Dim.2   Dim.3   Dim.4   Dim.5   Dim.6
## Variance               0.241   0.230   0.191   0.176   0.155   0.141
## % of var.             14.073  13.405  11.145  10.285   9.038   8.235
## Cumulative % of var.  14.073  27.477  38.622  48.907  57.945  66.180
##                        Dim.7   Dim.8   Dim.9  Dim.10  Dim.11  Dim.12
## Variance               0.126   0.121   0.116   0.090   0.074   0.053
## % of var.              7.337   7.069   6.790   5.221   4.321   3.082
## Cumulative % of var.  73.517  80.586  87.376  92.597  96.918 100.000
## 
## Individuals (the 10 first)
##                       Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3
## 1                  | -0.102  0.014  0.010 | -0.498  0.359  0.227 | -0.095
## 2                  | -0.227  0.071  0.036 | -0.107  0.017  0.008 | -0.557
## 3                  | -0.356  0.175  0.211 | -0.206  0.061  0.070 | -0.382
## 4                  | -0.319  0.141  0.139 | -0.534  0.413  0.390 |  0.285
## 5                  | -0.223  0.069  0.070 | -0.403  0.236  0.228 | -0.172
## 6                  | -0.223  0.069  0.070 | -0.403  0.236  0.228 | -0.172
## 7                  | -0.223  0.069  0.070 | -0.403  0.236  0.228 | -0.172
## 8                  | -0.227  0.071  0.036 | -0.107  0.017  0.008 | -0.557
## 9                  |  0.028  0.001  0.000 |  0.667  0.646  0.262 |  0.198
## 10                 |  0.334  0.154  0.070 |  0.641  0.597  0.257 | -0.384
##                       ctr   cos2  
## 1                   0.016  0.008 |
## 2                   0.542  0.219 |
## 3                   0.254  0.242 |
## 4                   0.141  0.111 |
## 5                   0.052  0.042 |
## 6                   0.052  0.042 |
## 7                   0.052  0.042 |
## 8                   0.542  0.219 |
## 9                   0.069  0.023 |
## 10                  0.258  0.092 |
## 
## Categories (the 10 first)
##                       Dim.1    ctr   cos2 v.test    Dim.2    ctr   cos2
## black              |  0.459  3.076  0.069  4.541 |  0.176  0.474  0.010
## Earl Grey          | -0.289  3.177  0.150 -6.706 |  0.054  0.118  0.005
## green              |  0.660  2.836  0.054  4.011 | -0.712  3.469  0.063
## alone              |  0.003  0.000  0.000  0.068 | -0.182  1.344  0.062
## lemon              |  0.588  2.252  0.043  3.575 |  0.549  2.063  0.037
## milk               | -0.301  1.128  0.024 -2.685 |  0.027  0.009  0.000
## other              | -0.111  0.022  0.000 -0.337 |  1.751  5.716  0.095
## tea bag            | -0.503  8.475  0.330 -9.937 | -0.491  8.501  0.316
## tea bag+unpackaged |  0.091  0.153  0.004  1.060 |  1.076 22.557  0.528
## unpackaged         |  2.136 32.429  0.622 13.641 | -0.490  1.792  0.033
##                    v.test    Dim.3    ctr   cos2 v.test  
## black               1.739 | -0.788 11.452  0.203 -7.797 |
## Earl Grey           1.263 |  0.372  6.662  0.250  8.642 |
## green              -4.330 | -0.409  1.379  0.021 -2.489 |
## alone              -4.298 | -0.275  3.682  0.141 -6.486 |
## lemon               3.339 |  1.384 15.750  0.237  8.412 |
## milk                0.239 |  0.347  1.891  0.032  3.094 |
## other               5.324 | -1.539  5.313  0.073 -4.680 |
## tea bag            -9.714 | -0.053  0.119  0.004 -1.047 |
## tea bag+unpackaged 12.570 |  0.075  0.132  0.003  0.877 |
## unpackaged         -3.129 |  0.054  0.026  0.000  0.345 |
## 
## Categorical variables (eta2)
##                      Dim.1 Dim.2 Dim.3  
## Tea                | 0.153 0.065 0.261 |
## How                | 0.057 0.147 0.356 |
## how                | 0.693 0.528 0.004 |
## sugar              | 0.027 0.048 0.487 |
## where              | 0.704 0.657 0.044 |
## sex                | 0.050 0.106 0.099 |
## lunch              | 0.003 0.057 0.086 |
# visualize MCA
plot(mca, invisible=c("ind"), habillage = "quali")

With our chosen variables: MCA plot allows us to observe that unpacked tea is bought from actual tea shops whereas tea bags are more likely to be bough from chain stores. We can also see that use of sugar is more common with tea bags than with unpacked tea. We can also see that tea Earl Grey is grouped closely to the use of milk in tea. Black tea is grouped closer to no sugar and use of lemon in tea.